1165:
209:
40:
2136:
1720:
2866:
724:
4127:
20:
3419:
2493:
2131:{\displaystyle K={\frac {{\begin{vmatrix}-{\frac {1}{2}}E_{vv}+F_{uv}-{\frac {1}{2}}G_{uu}&{\frac {1}{2}}E_{u}&F_{u}-{\frac {1}{2}}E_{v}\\F_{v}-{\frac {1}{2}}G_{u}&E&F\\{\frac {1}{2}}G_{v}&F&G\end{vmatrix}}-{\begin{vmatrix}0&{\frac {1}{2}}E_{v}&{\frac {1}{2}}G_{u}\\{\frac {1}{2}}E_{v}&E&F\\{\frac {1}{2}}G_{u}&F&G\end{vmatrix}}}{\left(EG-F^{2}\right)^{2}}}}
3183:
2861:{\displaystyle K=-{\frac {\begin{vmatrix}H(F)&\nabla F^{\mathsf {T}}\\\nabla F&0\end{vmatrix}}{|\nabla F|^{4}}}=-{\frac {\begin{vmatrix}F_{xx}&F_{xy}&F_{xz}&F_{x}\\F_{xy}&F_{yy}&F_{yz}&F_{y}\\F_{xz}&F_{yz}&F_{zz}&F_{z}\\F_{x}&F_{y}&F_{z}&0\\\end{vmatrix}}{|\nabla F|^{4}}}}
2290:
332:. At such points, the surface will be saddle shaped. Because one principal curvature is negative, one is positive, and the normal curvature varies continuously if you rotate a plane orthogonal to the surface around the normal to the surface in two directions, the normal curvatures will be zero giving the
1552:
There are many other possible bounded surfaces with constant
Gaussian curvature. Whilst the sphere is rigid and can not be bent using an isometry, if a small region removed, or even a cut along a small segment, then the resulting surface can be bent. Such bending preserves Gaussian curvature so any
593:
3414:{\displaystyle K=-{\frac {1}{E}}\left({\frac {\partial }{\partial u}}\Gamma _{12}^{2}-{\frac {\partial }{\partial v}}\Gamma _{11}^{2}+\Gamma _{12}^{1}\Gamma _{11}^{2}-\Gamma _{11}^{1}\Gamma _{12}^{2}+\Gamma _{12}^{2}\Gamma _{12}^{2}-\Gamma _{11}^{2}\Gamma _{22}^{2}\right)}
2157:
1701:
2446:
309:, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points, the surface will be dome like, locally lying on one side of its tangent plane. All sectional curvatures will have the same sign.
468:
457:(being the product of the eigenvalues of the Hessian). (Recall that the Hessian is the 2Ă2 matrix of second derivatives.) This definition allows one immediately to grasp the distinction between a cup/cap versus a saddle point.
1513:
1154:
3170:
3060:
841:
1035:
and a flat plane has constant curvature 0, these two surfaces are not isometric, not even locally. Thus any planar representation of even a small part of a sphere must distort the distances. Therefore, no
872:(Latin: "remarkable theorem") states that Gaussian curvature of a surface can be determined from the measurements of length on the surface itself. In fact, it can be found given the full knowledge of the
2954:
1367:
359:
Most surfaces will contain regions of positive
Gaussian curvature (elliptical points) and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a
694:
1183:
are locally isometric. A consequence of
Minding's theorem is that any surface whose curvature is identically zero can be constructed by bending some plane region. Such surfaces are called
1591:
132:
2316:
2285:{\displaystyle K=-{\frac {1}{2{\sqrt {EG}}}}\left({\frac {\partial }{\partial u}}{\frac {G_{u}}{\sqrt {EG}}}+{\frac {\partial }{\partial v}}{\frac {E_{v}}{\sqrt {EG}}}\right).}
1411:
1543:
3077:
1545:
either have a boundary or a singular point. do Carmo also gives three different examples of surface with constant negative
Gaussian curvature, one of which is
241:. For most points on most âsmoothâ surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the
2971:
766:
158:
everywhere, and a flat plane and a cylinder have
Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a
588:{\displaystyle K={\frac {{\bigl \langle }(\nabla _{2}\nabla _{1}-\nabla _{1}\nabla _{2})\mathbf {e} _{1},\mathbf {e} _{2}{\bigr \rangle }}{\det g}},}
2902:
1064:
4037:
429:
at the point. They measure how the surface bends by different amounts in different directions from that point. We represent the surface by the
1228:
908:
certainly depends on the way in which the surface is located in space, the end result, the
Gaussian curvature itself, is determined by the
645:
1416:
4163:
1516:
3691:
1277:
91:
3760:
3741:
3571:
3816:
4030:
3863:
1168:
Two surfaces which both have constant positive
Gaussian curvature but with either an open boundary or singular points.
3951:
3888:
3712:
3607:
3543:
3513:
3488:
935:
1266:
has constant negative
Gaussian curvature except at its boundary circle, where the gaussian curvature is not defined.
177:, depending only on distances that are measured âwithinâ or along the surface, not on the way it is isometrically
3838:
445:
vanishes (this can always be attained by a suitable rigid motion). Then the
Gaussian curvature of the surface at
3794:
4173:
4158:
920:
1216:. If a sphere is deformed, it does not remain a sphere, proving that a sphere is rigid. A standard proof uses
4178:
4131:
4023:
3787:
727:
The sum of the angles of a triangle on a surface of negative curvature is less than that of a plane triangle.
4112:
3918:
3782:
3529:
3777:
3430:
4153:
4102:
3858:
1372:
1058:
and provides an important link between local geometric properties and global topological properties.
430:
3651:
894:
can be so expressed. The "remarkable", and surprising, feature of this theorem is that although the
3981:
3956:
3878:
3450:
1696:{\displaystyle K={\frac {\det(\mathrm {I\!I} )}{\det(\mathrm {I} )}}={\frac {LN-M^{2}}{EG-F^{2}}}.}
1164:
1049:
844:
1244:
of constant negative
Gaussian curvature. In fact, the conclusion also holds for surfaces of class
43:
Some points on the torus have positive, some have negative, and some have zero Gaussian curvature.
3928:
3809:
3623:
2441:{\displaystyle K={\frac {F_{xx}\cdot F_{yy}-F_{xy}^{2}}{\left(1+F_{x}^{2}+F_{y}^{2}\right)^{2}}}}
1573:
1037:
931:
916:
885:
751:, while the sum of the angles of a triangle on a surface of negative curvature will be less than
398:
225:
3933:
3923:
3646:
2142:
1577:
873:
712:
390:
and patches of spheres have this geometry, but there exist other examples as well, such as the
382:
When a surface has a constant positive Gaussian curvature, then the geometry of the surface is
3533:
1021:
tube is zero, the same as for the "unrolled" tube (which is flat). On the other hand, since a
4057:
3830:
1522:
927:
48:
4046:
1553:
such bending of a sphere with a region removed will also have constant Gaussian curvature.
1055:
616:
191:
952:
given by the first fundamental form. Suppose that the image of the embedding is a surface
328:, then the Gaussian curvature is negative and the surface is said to have a hyperbolic or
8:
3971:
3943:
3898:
3455:
3435:
3176:
1184:
1018:
708:
402:
372:
242:
67:
28:
39:
3853:
3802:
3596:
3591:
3477:
1714:
1217:
877:
418:
383:
376:
223:
that is at right angles to the surface; planes containing the normal vector are called
63:
208:
4168:
4077:
4072:
3756:
3737:
3708:
3603:
3567:
3539:
3509:
3484:
1271:
1195:
1174:
949:
863:
747:. The sum of the angles of a triangle on a surface of positive curvature will exceed
740:
220:
196:
183:
4092:
3966:
3868:
3656:
2896:
909:
732:
391:
355:, the Gaussian curvature is zero and the surface is said to have a parabolic point.
333:
237:
4087:
3961:
3873:
3824:
2964:
1221:
756:
229:. The intersection of a normal plane and the surface will form a curve called a
4107:
4097:
4067:
3994:
3989:
3908:
3660:
3440:
2477:
1188:
967:
701:
450:
426:
360:
231:
212:
4147:
3913:
3587:
3070:
2959:
624:
912:
of the surface without any further reference to the ambient space: it is an
289:
The sign of the Gaussian curvature can be used to characterise the surface.
3637:
Goldman, R. (2005). "Curvature formulas for implicit curves and surfaces".
1546:
1263:
1224:
points of extreme principal curvature have non-positive Gaussian curvature.
1149:{\displaystyle \int _{M}K\,dA+\int _{\partial M}k_{g}\,ds=2\pi \chi (M),\,}
329:
1717:) gives Gaussian curvature solely in terms of the first fundamental form:
1270:
There are other surfaces which have constant positive Gaussian curvature.
3999:
3165:{\displaystyle K=\lim _{r\to 0^{+}}12{\frac {\pi r^{2}-A(r)}{\pi r^{4}}}}
1569:
1054:
The GaussâBonnet theorem relates the total curvature of a surface to its
881:
422:
159:
24:
3561:
397:
When a surface has a constant negative Gaussian curvature, then it is a
4015:
1519:). These surfaces all have constant Gaussian curvature of 1, but, for
735:
of the Gaussian curvature over some region of a surface is called the
723:
4062:
3848:
3826:
3445:
3055:{\displaystyle K=\lim _{r\to 0^{+}}3{\frac {2\pi r-C(r)}{\pi r^{3}}}}
939:
913:
178:
174:
19:
3535:
Modern Differential Geometry of Curves and Surfaces with Mathematica
1508:{\textstyle \psi (v)=\int _{0}^{v}{\sqrt {1-C^{2}\sin ^{2}v'}}\ dv'}
371:
When a surface has a constant zero Gaussian curvature, then it is a
4004:
3734:
Introduction to Tensor Analysis and the Calculus of Moving Surfaces
2469:, the Gaussian curvature can be expressed in terms of the gradient
3586:
1179:(1839) states that all surfaces with the same constant curvature
836:{\displaystyle \sum _{i=1}^{3}\theta _{i}=\pi +\iint _{T}K\,dA.}
1213:
1200:(1900) answered Minding's question. The only regular (of class
1022:
387:
32:
3566:(2nd ed.). Mineola, NY: Dover Publications. p. 171.
23:
From left to right: a surface of negative Gaussian curvature (
2870:
For a surface with metric conformal to the Euclidean one, so
163:
3624:"General investigations of curved surfaces of 1827 and 1825"
1232:(1901) states that there exists no complete analytic (class
66:
in three-dimensional space at a point is the product of the
3065:
2949:{\displaystyle K=-{\frac {1}{2e^{\sigma }}}\Delta \sigma .}
919:. In particular, the Gaussian curvature is invariant under
3064:
Gaussian curvature is the limiting difference between the
2958:
Gaussian curvature is the limiting difference between the
1362:{\displaystyle (\phi (v)\cos(u),\phi (v)\sin(u),\psi (v))}
1007:
The Gaussian curvature of an embedded smooth surface in
215:
with normal planes in directions of principal curvatures
3532:(1997). "28.4 Hilbert's Lemma and Liebmann's Theorem".
930:, a "surface", viewed abstractly, is a two-dimensional
2605:
2512:
1963:
1738:
1419:
1191:
with constant positive curvature is necessarily rigid.
3676:
A Comprehensive Introduction to Differential Geometry
3186:
3080:
2974:
2905:
2496:
2319:
2160:
1723:
1594:
1525:
1375:
1280:
1067:
876:
and expressed via the first fundamental form and its
769:
689:{\displaystyle K(\mathbf {p} )=\det S(\mathbf {p} ),}
648:
471:
94:
743:equals the deviation of the sum of its angles from
408:
293:If both principal curvatures are of the same sign:
3595:
3506:Differential Geometry: Curves, Surfaces, Manifolds
3476:
3413:
3164:
3054:
2948:
2860:
2440:
2284:
2130:
1695:
1537:
1507:
1405:
1361:
1148:
835:
688:
587:
312:If the principal curvatures have different signs:
126:
3687:
3685:
1614:
1159:
437:, of two variables, in such a way that the point
31:), and a surface of positive Gaussian curvature (
4145:
3088:
2982:
1624:
1604:
1187:. Minding also raised the question of whether a
666:
573:
16:Product of the principal curvatures of a surface
2294:For a surface described as graph of a function
1517:incomplete Elliptic integral of the second kind
707:A useful formula for the Gaussian curvature is
267:is the product of the two principal curvatures
181:in Euclidean space. This is the content of the
3682:
1212:with constant positive Gaussian curvature are
755:. On a surface of zero curvature, such as the
441:is a critical point, that is, the gradient of
4031:
3810:
3538:(2nd ed.). CRC Press. pp. 652â654.
3174:Gaussian curvature may be expressed with the
880:of first and second order. Equivalently, the
566:
483:
3563:Differential geometry of curves and surfaces
339:If one of the principal curvatures is zero:
3705:Lectures on Classical Differential Geometry
4038:
4024:
3817:
3803:
3626:. The Princeton university library. 1902.
642:, the Gaussian curvature is also given by
460:
3678:. Vol. 3. Boston: Publish or Perish.
3650:
1145:
1114:
1081:
1017:For example, the Gaussian curvature of a
1013:is invariant under the local isometries.
934:. To connect this point of view with the
823:
366:
219:At any point on a surface, we can find a
127:{\displaystyle K=\kappa _{1}\kappa _{2}.}
27:), a surface of zero Gaussian curvature (
4045:
3731:
3474:
1163:
722:
207:
38:
18:
3750:
3636:
1556:
1043:
898:of the Gaussian curvature of a surface
235:and the curvature of this curve is the
4146:
3702:
3673:
3602:(2nd ed.). Chelsea. p. 228.
3503:
2538:
203:
4019:
3798:
3560:Carmo, Manfredo PerdigĂŁo do (2016) .
3559:
1568:can be expressed as the ratio of the
850:
3528:
3431:Earth's Gaussian radius of curvature
855:
2891:, the Gauss curvature is given by (
2450:For an implicitly defined surface,
1562:Gaussian curvature of a surface in
759:, the angles will sum to precisely
401:and the geometry of the surface is
375:and the geometry of the surface is
13:
3864:Radius of curvature (applications)
3392:
3377:
3359:
3344:
3326:
3311:
3293:
3278:
3260:
3250:
3246:
3227:
3217:
3213:
2937:
2837:
2572:
2548:
2529:
2243:
2239:
2203:
2199:
1631:
1615:
1611:
1096:
718:
525:
515:
502:
492:
190:Gaussian curvature is named after
14:
4190:
4164:Differential geometry of surfaces
3952:Curvature of Riemannian manifolds
3770:
3753:General Relativity the Essentials
3508:. American Mathematical Society.
1274:considers surfaces of revolution
4126:
4125:
1406:{\displaystyle \phi (v)=C\cos v}
1029:has constant positive curvature
676:
656:
554:
539:
409:Relation to principal curvatures
134:For example, a sphere of radius
3696:
3692:BertrandâDiquetâPuiseux theorem
3639:Computer Aided Geometric Design
999:is an isometry onto its image.
3755:. Cambridge University Press.
3707:. Courier Dover Publications.
3667:
3630:
3616:
3580:
3553:
3522:
3497:
3483:. Cambridge University Press.
3468:
3141:
3135:
3095:
3031:
3025:
2989:
2845:
2833:
2580:
2568:
2524:
2518:
1635:
1627:
1619:
1607:
1429:
1423:
1385:
1379:
1356:
1353:
1347:
1338:
1332:
1323:
1317:
1308:
1302:
1293:
1287:
1281:
1160:Surfaces of constant curvature
1139:
1133:
938:, such an abstract surface is
680:
672:
660:
652:
534:
488:
1:
3461:
923:deformations of the surface.
843:A more general result is the
711:in terms of the Laplacian in
3598:Geometry and the Imagination
936:classical theory of surfaces
433:as the graph of a function,
7:
3783:Encyclopedia of Mathematics
3424:
2968:and a circle in the plane:
1003:is then stated as follows:
739:. The total curvature of a
10:
4195:
3661:10.1016/j.cagd.2005.06.005
2154:), Gaussian curvature is:
1047:
861:
449:is the determinant of the
4121:
4113:Gauss's law for magnetism
4053:
3980:
3942:
3887:
3837:
3504:KĂŒhnel, Wolfgang (2006).
3479:Geometric Differentiation
3074:and a disk in the plane:
2313:, Gaussian curvature is:
431:implicit function theorem
392:lemon / American football
169:Gaussian curvature is an
3982:Curvature of connections
3957:Riemann curvature tensor
3879:Total absolute curvature
3725:
3475:Porteous, I. R. (1994).
3451:Riemann curvature tensor
983:between open regions of
636:on a regular surface in
4103:Gauss's law for gravity
3929:Second fundamental form
3919:GaussâCodazzi equations
3751:Rovelli, Carlo (2021).
1538:{\displaystyle C\neq 1}
1038:cartographic projection
932:differentiable manifold
886:second fundamental form
461:Alternative definitions
399:pseudospherical surface
138:has Gaussian curvature
3934:Third fundamental form
3924:First fundamental form
3889:Differential geometry
3859:FrenetâSerret formulas
3839:Differential geometry
3415:
3166:
3056:
2950:
2862:
2442:
2286:
2132:
1697:
1539:
1509:
1407:
1363:
1256:, but breaks down for
1169:
1150:
874:first fundamental form
837:
790:
728:
713:isothermal coordinates
690:
589:
417:at a given point of a
367:Relation to geometries
216:
128:
88:, at the given point:
44:
36:
4174:Differential topology
4159:Differential geometry
4058:Gauss composition law
3831:differential geometry
3732:Grinfeld, P. (2014).
3703:Struik, Dirk (1988).
3416:
3167:
3057:
2951:
2863:
2443:
2287:
2133:
1698:
1540:
1510:
1408:
1364:
1238:) regular surface in
1206:) closed surfaces in
1167:
1151:
989:whose restriction to
948:and endowed with the
928:differential geometry
838:
770:
726:
691:
590:
211:
129:
49:differential geometry
42:
22:
4179:Carl Friedrich Gauss
4047:Carl Friedrich Gauss
3899:Principal curvatures
3778:"Gaussian curvature"
3592:Cohn-Vossen, Stephan
3184:
3078:
2972:
2903:
2494:
2317:
2158:
1721:
1592:
1557:Alternative formulas
1523:
1417:
1373:
1278:
1185:developable surfaces
1065:
1056:Euler characteristic
1050:GaussâBonnet theorem
1044:GaussâBonnet theorem
845:GaussâBonnet theorem
767:
709:Liouville's equation
646:
617:covariant derivative
469:
465:It is also given by
415:principal curvatures
243:principal curvatures
194:, who published the
192:Carl Friedrich Gauss
92:
68:principal curvatures
3972:Sectional curvature
3944:Riemannian geometry
3825:Various notions of
3674:Spivak, M. (1975).
3576:– via zbMATH.
3456:Principal curvature
3436:Sectional curvature
3405:
3390:
3372:
3357:
3339:
3324:
3306:
3291:
3273:
3240:
3177:Christoffel symbols
2424:
2406:
2378:
1449:
878:partial derivatives
403:hyperbolic geometry
373:developable surface
204:Informal definition
162:or the inside of a
4083:Gaussian curvature
3904:Gaussian curvature
3854:Torsion of a curve
3411:
3391:
3376:
3358:
3343:
3325:
3310:
3292:
3277:
3259:
3226:
3162:
3109:
3052:
3003:
2946:
2858:
2826:
2561:
2438:
2410:
2392:
2361:
2282:
2128:
2083:
1949:
1715:Francesco Brioschi
1693:
1580:fundamental forms
1535:
1505:
1435:
1403:
1359:
1170:
1146:
851:Important theorems
833:
729:
686:
585:
384:spherical geometry
377:Euclidean geometry
265:Gaussian curvature
217:
124:
53:Gaussian curvature
45:
37:
4154:Curvature tensors
4141:
4140:
4078:Gaussian brackets
4013:
4012:
3762:978-1-009-01369-7
3743:978-1-4614-7866-9
3573:978-0-486-80699-0
3257:
3224:
3204:
3160:
3087:
3050:
2981:
2935:
2856:
2591:
2436:
2272:
2271:
2250:
2232:
2231:
2210:
2190:
2187:
2126:
2059:
2025:
2001:
1979:
1925:
1891:
1854:
1819:
1794:
1752:
1688:
1639:
1493:
1489:
1272:Manfredo do Carmo
1229:Hilbert's theorem
1001:Theorema egregium
950:Riemannian metric
870:Theorema egregium
864:Theorema egregium
857:Theorema egregium
741:geodesic triangle
580:
334:asymptotic curves
197:Theorema egregium
184:Theorema egregium
4186:
4129:
4128:
4093:Gaussian surface
4040:
4033:
4026:
4017:
4016:
3967:Scalar curvature
3869:Affine curvature
3819:
3812:
3805:
3796:
3795:
3791:
3766:
3747:
3719:
3718:
3700:
3694:
3689:
3680:
3679:
3671:
3665:
3664:
3654:
3634:
3628:
3627:
3620:
3614:
3613:
3601:
3584:
3578:
3577:
3557:
3551:
3549:
3526:
3520:
3519:
3501:
3495:
3494:
3482:
3472:
3420:
3418:
3417:
3412:
3410:
3406:
3404:
3399:
3389:
3384:
3371:
3366:
3356:
3351:
3338:
3333:
3323:
3318:
3305:
3300:
3290:
3285:
3272:
3267:
3258:
3256:
3245:
3239:
3234:
3225:
3223:
3212:
3205:
3197:
3171:
3169:
3168:
3163:
3161:
3159:
3158:
3157:
3144:
3128:
3127:
3114:
3108:
3107:
3106:
3061:
3059:
3058:
3053:
3051:
3049:
3048:
3047:
3034:
3008:
3002:
3001:
3000:
2955:
2953:
2952:
2947:
2936:
2934:
2933:
2932:
2916:
2897:Laplace operator
2895:being the usual
2894:
2890:
2876:
2867:
2865:
2864:
2859:
2857:
2855:
2854:
2853:
2848:
2836:
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2779:
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2767:
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2694:
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2600:
2592:
2590:
2589:
2588:
2583:
2571:
2565:
2543:
2542:
2541:
2507:
2489:
2475:
2468:
2447:
2445:
2444:
2439:
2437:
2435:
2434:
2429:
2425:
2423:
2418:
2405:
2400:
2379:
2377:
2372:
2357:
2356:
2341:
2340:
2327:
2312:
2291:
2289:
2288:
2283:
2278:
2274:
2273:
2264:
2263:
2262:
2253:
2251:
2249:
2238:
2233:
2224:
2223:
2222:
2213:
2211:
2209:
2198:
2191:
2189:
2188:
2180:
2171:
2153:
2137:
2135:
2134:
2129:
2127:
2125:
2124:
2119:
2115:
2114:
2113:
2089:
2088:
2087:
2070:
2069:
2060:
2052:
2036:
2035:
2026:
2018:
2012:
2011:
2002:
1994:
1990:
1989:
1980:
1972:
1954:
1953:
1936:
1935:
1926:
1918:
1902:
1901:
1892:
1884:
1879:
1878:
1865:
1864:
1855:
1847:
1842:
1841:
1830:
1829:
1820:
1812:
1808:
1807:
1795:
1787:
1782:
1781:
1766:
1765:
1753:
1745:
1731:
1711:
1710:
1709:Brioschi formula
1702:
1700:
1699:
1694:
1689:
1687:
1686:
1685:
1666:
1665:
1664:
1645:
1640:
1638:
1634:
1622:
1618:
1602:
1587:
1583:
1567:
1544:
1542:
1541:
1536:
1514:
1512:
1511:
1506:
1504:
1491:
1490:
1488:
1477:
1476:
1467:
1466:
1451:
1448:
1443:
1412:
1410:
1409:
1404:
1368:
1366:
1365:
1360:
1261:
1255:
1249:
1243:
1237:
1211:
1205:
1182:
1155:
1153:
1152:
1147:
1113:
1112:
1103:
1102:
1077:
1076:
1034:
1028:
1012:
998:
988:
982:
961:
955:
947:
926:In contemporary
910:intrinsic metric
907:
901:
893:
888:of a surface in
842:
840:
839:
834:
819:
818:
800:
799:
789:
784:
762:
754:
750:
746:
733:surface integral
699:
695:
693:
692:
687:
679:
659:
641:
635:
622:
614:
594:
592:
591:
586:
581:
579:
571:
570:
569:
563:
562:
557:
548:
547:
542:
533:
532:
523:
522:
510:
509:
500:
499:
487:
486:
479:
456:
448:
444:
440:
436:
354:
327:
308:
285:
262:
253:
238:normal curvature
157:
156:
154:
153:
148:
145:
137:
133:
131:
130:
125:
120:
119:
110:
109:
87:
78:
61:
4194:
4193:
4189:
4188:
4187:
4185:
4184:
4183:
4144:
4143:
4142:
4137:
4117:
4088:Gaussian period
4049:
4044:
4014:
4009:
3976:
3962:Ricci curvature
3938:
3890:
3883:
3874:Total curvature
3840:
3833:
3823:
3776:
3773:
3763:
3744:
3728:
3723:
3722:
3715:
3701:
3697:
3690:
3683:
3672:
3668:
3652:10.1.1.413.3008
3635:
3631:
3622:
3621:
3617:
3610:
3585:
3581:
3574:
3558:
3554:
3546:
3527:
3523:
3516:
3502:
3498:
3491:
3473:
3469:
3464:
3427:
3400:
3395:
3385:
3380:
3367:
3362:
3352:
3347:
3334:
3329:
3319:
3314:
3301:
3296:
3286:
3281:
3268:
3263:
3249:
3244:
3235:
3230:
3216:
3211:
3210:
3206:
3196:
3185:
3182:
3181:
3153:
3149:
3145:
3123:
3119:
3115:
3113:
3102:
3098:
3091:
3079:
3076:
3075:
3043:
3039:
3035:
3009:
3007:
2996:
2992:
2985:
2973:
2970:
2969:
2965:geodesic circle
2928:
2924:
2920:
2915:
2904:
2901:
2900:
2892:
2878:
2871:
2849:
2844:
2843:
2832:
2831:
2825:
2824:
2819:
2813:
2809:
2807:
2801:
2797:
2795:
2789:
2785:
2782:
2781:
2775:
2771:
2769:
2760:
2756:
2754:
2745:
2741:
2739:
2730:
2726:
2723:
2722:
2716:
2712:
2710:
2701:
2697:
2695:
2686:
2682:
2680:
2671:
2667:
2664:
2663:
2657:
2653:
2651:
2642:
2638:
2636:
2627:
2623:
2621:
2612:
2608:
2601:
2599:
2584:
2579:
2578:
2567:
2566:
2560:
2559:
2554:
2545:
2544:
2537:
2536:
2532:
2527:
2508:
2506:
2495:
2492:
2491:
2480:
2470:
2451:
2430:
2419:
2414:
2401:
2396:
2385:
2381:
2380:
2373:
2365:
2349:
2345:
2333:
2329:
2328:
2326:
2318:
2315:
2314:
2295:
2258:
2254:
2252:
2242:
2237:
2218:
2214:
2212:
2202:
2197:
2196:
2192:
2179:
2175:
2170:
2159:
2156:
2155:
2148:
2145:parametrization
2120:
2109:
2105:
2095:
2091:
2090:
2082:
2081:
2076:
2071:
2065:
2061:
2051:
2048:
2047:
2042:
2037:
2031:
2027:
2017:
2014:
2013:
2007:
2003:
1993:
1991:
1985:
1981:
1971:
1969:
1959:
1958:
1948:
1947:
1942:
1937:
1931:
1927:
1917:
1914:
1913:
1908:
1903:
1897:
1893:
1883:
1874:
1870:
1867:
1866:
1860:
1856:
1846:
1837:
1833:
1831:
1825:
1821:
1811:
1809:
1800:
1796:
1786:
1774:
1770:
1758:
1754:
1744:
1734:
1733:
1732:
1730:
1722:
1719:
1718:
1708:
1707:
1681:
1677:
1667:
1660:
1656:
1646:
1644:
1630:
1623:
1610:
1603:
1601:
1593:
1590:
1589:
1585:
1581:
1563:
1559:
1524:
1521:
1520:
1497:
1481:
1472:
1468:
1462:
1458:
1450:
1444:
1439:
1418:
1415:
1414:
1374:
1371:
1370:
1279:
1276:
1275:
1262:-surfaces. The
1257:
1251:
1245:
1239:
1233:
1218:Hilbert's lemma
1207:
1201:
1180:
1162:
1108:
1104:
1095:
1091:
1072:
1068:
1066:
1063:
1062:
1052:
1046:
1030:
1026:
1015:
1008:
990:
984:
970:
957:
953:
943:
903:
899:
889:
866:
860:
853:
814:
810:
795:
791:
785:
774:
768:
765:
764:
760:
757:Euclidean plane
752:
748:
744:
737:total curvature
721:
719:Total curvature
697:
675:
655:
647:
644:
643:
637:
631:
620:
613:
612:
602:
596:
572:
565:
564:
558:
553:
552:
543:
538:
537:
528:
524:
518:
514:
505:
501:
495:
491:
482:
481:
480:
478:
470:
467:
466:
463:
454:
446:
442:
438:
434:
411:
369:
352:
346:
340:
336:for that point.
325:
319:
313:
306:
300:
294:
284:
278:
268:
261:
255:
252:
246:
206:
149:
146:
143:
142:
140:
139:
135:
115:
111:
105:
101:
93:
90:
89:
86:
80:
77:
71:
59:
57:Gauss curvature
17:
12:
11:
5:
4192:
4182:
4181:
4176:
4171:
4166:
4161:
4156:
4139:
4138:
4136:
4135:
4122:
4119:
4118:
4116:
4115:
4110:
4105:
4100:
4098:Gaussian units
4095:
4090:
4085:
4080:
4075:
4073:Gauss's method
4070:
4068:Gauss notation
4065:
4060:
4054:
4051:
4050:
4043:
4042:
4035:
4028:
4020:
4011:
4010:
4008:
4007:
4002:
3997:
3995:Torsion tensor
3992:
3990:Curvature form
3986:
3984:
3978:
3977:
3975:
3974:
3969:
3964:
3959:
3954:
3948:
3946:
3940:
3939:
3937:
3936:
3931:
3926:
3921:
3916:
3911:
3909:Mean curvature
3906:
3901:
3895:
3893:
3885:
3884:
3882:
3881:
3876:
3871:
3866:
3861:
3856:
3851:
3845:
3843:
3835:
3834:
3822:
3821:
3814:
3807:
3799:
3793:
3792:
3772:
3771:External links
3769:
3768:
3767:
3761:
3748:
3742:
3727:
3724:
3721:
3720:
3713:
3695:
3681:
3666:
3645:(7): 632â658.
3629:
3615:
3608:
3588:Hilbert, David
3579:
3572:
3552:
3544:
3521:
3514:
3496:
3489:
3466:
3465:
3463:
3460:
3459:
3458:
3453:
3448:
3443:
3441:Mean curvature
3438:
3433:
3426:
3423:
3422:
3421:
3409:
3403:
3398:
3394:
3388:
3383:
3379:
3375:
3370:
3365:
3361:
3355:
3350:
3346:
3342:
3337:
3332:
3328:
3322:
3317:
3313:
3309:
3304:
3299:
3295:
3289:
3284:
3280:
3276:
3271:
3266:
3262:
3255:
3252:
3248:
3243:
3238:
3233:
3229:
3222:
3219:
3215:
3209:
3203:
3200:
3195:
3192:
3189:
3172:
3156:
3152:
3148:
3143:
3140:
3137:
3134:
3131:
3126:
3122:
3118:
3112:
3105:
3101:
3097:
3094:
3090:
3086:
3083:
3062:
3046:
3042:
3038:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3006:
2999:
2995:
2991:
2988:
2984:
2980:
2977:
2956:
2945:
2942:
2939:
2931:
2927:
2923:
2919:
2914:
2911:
2908:
2868:
2852:
2847:
2842:
2839:
2835:
2829:
2823:
2820:
2816:
2812:
2808:
2804:
2800:
2796:
2792:
2788:
2784:
2783:
2778:
2774:
2770:
2766:
2763:
2759:
2755:
2751:
2748:
2744:
2740:
2736:
2733:
2729:
2725:
2724:
2719:
2715:
2711:
2707:
2704:
2700:
2696:
2692:
2689:
2685:
2681:
2677:
2674:
2670:
2666:
2665:
2660:
2656:
2652:
2648:
2645:
2641:
2637:
2633:
2630:
2626:
2622:
2618:
2615:
2611:
2607:
2606:
2604:
2598:
2595:
2587:
2582:
2577:
2574:
2570:
2564:
2558:
2555:
2553:
2550:
2547:
2546:
2540:
2535:
2531:
2528:
2526:
2523:
2520:
2517:
2514:
2513:
2511:
2505:
2502:
2499:
2478:Hessian matrix
2448:
2433:
2428:
2422:
2417:
2413:
2409:
2404:
2399:
2395:
2391:
2388:
2384:
2376:
2371:
2368:
2364:
2360:
2355:
2352:
2348:
2344:
2339:
2336:
2332:
2325:
2322:
2292:
2281:
2277:
2270:
2267:
2261:
2257:
2248:
2245:
2241:
2236:
2230:
2227:
2221:
2217:
2208:
2205:
2201:
2195:
2186:
2183:
2178:
2174:
2169:
2166:
2163:
2138:
2123:
2118:
2112:
2108:
2104:
2101:
2098:
2094:
2086:
2080:
2077:
2075:
2072:
2068:
2064:
2058:
2055:
2050:
2049:
2046:
2043:
2041:
2038:
2034:
2030:
2024:
2021:
2016:
2015:
2010:
2006:
2000:
1997:
1992:
1988:
1984:
1978:
1975:
1970:
1968:
1965:
1964:
1962:
1957:
1952:
1946:
1943:
1941:
1938:
1934:
1930:
1924:
1921:
1916:
1915:
1912:
1909:
1907:
1904:
1900:
1896:
1890:
1887:
1882:
1877:
1873:
1869:
1868:
1863:
1859:
1853:
1850:
1845:
1840:
1836:
1832:
1828:
1824:
1818:
1815:
1810:
1806:
1803:
1799:
1793:
1790:
1785:
1780:
1777:
1773:
1769:
1764:
1761:
1757:
1751:
1748:
1743:
1740:
1739:
1737:
1729:
1726:
1703:
1692:
1684:
1680:
1676:
1673:
1670:
1663:
1659:
1655:
1652:
1649:
1643:
1637:
1633:
1629:
1626:
1621:
1617:
1613:
1609:
1606:
1600:
1597:
1558:
1555:
1534:
1531:
1528:
1503:
1500:
1496:
1487:
1484:
1480:
1475:
1471:
1465:
1461:
1457:
1454:
1447:
1442:
1438:
1434:
1431:
1428:
1425:
1422:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1268:
1267:
1225:
1192:
1189:closed surface
1161:
1158:
1157:
1156:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1111:
1107:
1101:
1098:
1094:
1090:
1087:
1084:
1080:
1075:
1071:
1048:Main article:
1045:
1042:
1005:
968:diffeomorphism
964:local isometry
862:Main article:
859:
854:
852:
849:
832:
829:
826:
822:
817:
813:
809:
806:
803:
798:
794:
788:
783:
780:
777:
773:
720:
717:
702:shape operator
685:
682:
678:
674:
671:
668:
665:
662:
658:
654:
651:
608:
604:
598:
584:
578:
575:
568:
561:
556:
551:
546:
541:
536:
531:
527:
521:
517:
513:
508:
504:
498:
494:
490:
485:
477:
474:
462:
459:
451:Hessian matrix
427:shape operator
410:
407:
368:
365:
361:parabolic line
357:
356:
350:
344:
337:
323:
317:
310:
304:
298:
282:
276:
259:
250:
232:normal section
213:Saddle surface
205:
202:
123:
118:
114:
108:
104:
100:
97:
84:
75:
15:
9:
6:
4:
3:
2:
4191:
4180:
4177:
4175:
4172:
4170:
4167:
4165:
4162:
4160:
4157:
4155:
4152:
4151:
4149:
4134:
4133:
4124:
4123:
4120:
4114:
4111:
4109:
4106:
4104:
4101:
4099:
4096:
4094:
4091:
4089:
4086:
4084:
4081:
4079:
4076:
4074:
4071:
4069:
4066:
4064:
4061:
4059:
4056:
4055:
4052:
4048:
4041:
4036:
4034:
4029:
4027:
4022:
4021:
4018:
4006:
4003:
4001:
3998:
3996:
3993:
3991:
3988:
3987:
3985:
3983:
3979:
3973:
3970:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3949:
3947:
3945:
3941:
3935:
3932:
3930:
3927:
3925:
3922:
3920:
3917:
3915:
3914:Darboux frame
3912:
3910:
3907:
3905:
3902:
3900:
3897:
3896:
3894:
3892:
3886:
3880:
3877:
3875:
3872:
3870:
3867:
3865:
3862:
3860:
3857:
3855:
3852:
3850:
3847:
3846:
3844:
3842:
3836:
3832:
3828:
3820:
3815:
3813:
3808:
3806:
3801:
3800:
3797:
3789:
3785:
3784:
3779:
3775:
3774:
3764:
3758:
3754:
3749:
3745:
3739:
3735:
3730:
3729:
3716:
3714:0-486-65609-8
3710:
3706:
3699:
3693:
3688:
3686:
3677:
3670:
3662:
3658:
3653:
3648:
3644:
3640:
3633:
3625:
3619:
3611:
3609:0-8284-1087-9
3605:
3600:
3599:
3593:
3589:
3583:
3575:
3569:
3565:
3564:
3556:
3547:
3545:9780849371646
3541:
3537:
3536:
3531:
3525:
3517:
3515:0-8218-3988-8
3511:
3507:
3500:
3492:
3490:0-521-39063-X
3486:
3481:
3480:
3471:
3467:
3457:
3454:
3452:
3449:
3447:
3444:
3442:
3439:
3437:
3434:
3432:
3429:
3428:
3407:
3401:
3396:
3386:
3381:
3373:
3368:
3363:
3353:
3348:
3340:
3335:
3330:
3320:
3315:
3307:
3302:
3297:
3287:
3282:
3274:
3269:
3264:
3253:
3241:
3236:
3231:
3220:
3207:
3201:
3198:
3193:
3190:
3187:
3179:
3178:
3173:
3154:
3150:
3146:
3138:
3132:
3129:
3124:
3120:
3116:
3110:
3103:
3099:
3092:
3084:
3081:
3073:
3072:
3071:geodesic disk
3067:
3063:
3044:
3040:
3036:
3028:
3022:
3019:
3016:
3013:
3010:
3004:
2997:
2993:
2986:
2978:
2975:
2967:
2966:
2961:
2960:circumference
2957:
2943:
2940:
2929:
2925:
2921:
2917:
2912:
2909:
2906:
2898:
2889:
2885:
2881:
2874:
2869:
2850:
2840:
2827:
2821:
2814:
2810:
2802:
2798:
2790:
2786:
2776:
2772:
2764:
2761:
2757:
2749:
2746:
2742:
2734:
2731:
2727:
2717:
2713:
2705:
2702:
2698:
2690:
2687:
2683:
2675:
2672:
2668:
2658:
2654:
2646:
2643:
2639:
2631:
2628:
2624:
2616:
2613:
2609:
2602:
2596:
2593:
2585:
2575:
2562:
2556:
2551:
2533:
2521:
2515:
2509:
2503:
2500:
2497:
2487:
2483:
2479:
2474:
2466:
2462:
2458:
2454:
2449:
2431:
2426:
2420:
2415:
2411:
2407:
2402:
2397:
2393:
2389:
2386:
2382:
2374:
2369:
2366:
2362:
2358:
2353:
2350:
2346:
2342:
2337:
2334:
2330:
2323:
2320:
2310:
2306:
2302:
2298:
2293:
2279:
2275:
2268:
2265:
2259:
2255:
2246:
2234:
2228:
2225:
2219:
2215:
2206:
2193:
2184:
2181:
2176:
2172:
2167:
2164:
2161:
2151:
2146:
2144:
2139:
2121:
2116:
2110:
2106:
2102:
2099:
2096:
2092:
2084:
2078:
2073:
2066:
2062:
2056:
2053:
2044:
2039:
2032:
2028:
2022:
2019:
2008:
2004:
1998:
1995:
1986:
1982:
1976:
1973:
1966:
1960:
1955:
1950:
1944:
1939:
1932:
1928:
1922:
1919:
1910:
1905:
1898:
1894:
1888:
1885:
1880:
1875:
1871:
1861:
1857:
1851:
1848:
1843:
1838:
1834:
1826:
1822:
1816:
1813:
1804:
1801:
1797:
1791:
1788:
1783:
1778:
1775:
1771:
1767:
1762:
1759:
1755:
1749:
1746:
1741:
1735:
1727:
1724:
1716:
1712:
1704:
1690:
1682:
1678:
1674:
1671:
1668:
1661:
1657:
1653:
1650:
1647:
1641:
1598:
1595:
1579:
1575:
1571:
1566:
1561:
1560:
1554:
1550:
1548:
1532:
1529:
1526:
1518:
1501:
1498:
1494:
1485:
1482:
1478:
1473:
1469:
1463:
1459:
1455:
1452:
1445:
1440:
1436:
1432:
1426:
1420:
1400:
1397:
1394:
1391:
1388:
1382:
1376:
1350:
1344:
1341:
1335:
1329:
1326:
1320:
1314:
1311:
1305:
1299:
1296:
1290:
1284:
1273:
1265:
1260:
1254:
1248:
1242:
1236:
1231:
1230:
1226:
1223:
1219:
1215:
1210:
1204:
1199:
1197:
1193:
1190:
1186:
1178:
1176:
1172:
1171:
1166:
1142:
1136:
1130:
1127:
1124:
1121:
1118:
1115:
1109:
1105:
1099:
1092:
1088:
1085:
1082:
1078:
1073:
1069:
1061:
1060:
1059:
1057:
1051:
1041:
1039:
1033:
1024:
1020:
1014:
1011:
1004:
1002:
997:
993:
987:
981:
977:
973:
969:
965:
960:
951:
946:
941:
937:
933:
929:
924:
922:
918:
915:
911:
906:
897:
892:
887:
883:
879:
875:
871:
865:
858:
848:
846:
830:
827:
824:
820:
815:
811:
807:
804:
801:
796:
792:
786:
781:
778:
775:
771:
758:
742:
738:
734:
725:
716:
714:
710:
705:
703:
683:
669:
663:
649:
640:
634:
628:
626:
625:metric tensor
618:
611:
607:
601:
582:
576:
559:
549:
544:
529:
519:
511:
506:
496:
475:
472:
458:
452:
432:
428:
424:
420:
416:
406:
404:
400:
395:
393:
389:
385:
380:
378:
374:
364:
362:
349:
343:
338:
335:
331:
322:
316:
311:
303:
297:
292:
291:
290:
287:
281:
275:
271:
266:
258:
249:
245:, call these
244:
240:
239:
234:
233:
228:
227:
226:normal planes
222:
221:normal vector
214:
210:
201:
199:
198:
193:
188:
186:
185:
180:
176:
172:
167:
165:
161:
152:
121:
116:
112:
106:
102:
98:
95:
83:
74:
69:
65:
58:
54:
50:
41:
34:
30:
26:
21:
4130:
4082:
3903:
3781:
3752:
3736:. Springer.
3733:
3704:
3698:
3675:
3669:
3642:
3638:
3632:
3618:
3597:
3582:
3562:
3555:
3534:
3530:Gray, Alfred
3524:
3505:
3499:
3478:
3470:
3175:
3069:
2963:
2887:
2883:
2879:
2872:
2485:
2481:
2472:
2464:
2460:
2456:
2452:
2308:
2304:
2300:
2296:
2149:
2141:
1706:
1570:determinants
1564:
1551:
1547:pseudosphere
1269:
1264:pseudosphere
1258:
1252:
1250:immersed in
1246:
1240:
1234:
1227:
1208:
1202:
1194:
1173:
1053:
1040:is perfect.
1031:
1016:
1009:
1006:
1000:
995:
991:
985:
979:
975:
971:
963:
958:
944:
925:
904:
895:
890:
869:
867:
856:
736:
730:
706:
638:
632:
629:
609:
605:
599:
464:
414:
412:
396:
381:
370:
358:
347:
341:
330:saddle point
320:
314:
301:
295:
288:
279:
273:
269:
264:
256:
247:
236:
230:
224:
218:
195:
189:
182:
170:
168:
150:
81:
72:
62:of a smooth
56:
52:
46:
4108:Gauss's law
4000:Cocurvature
3891:of surfaces
3829:defined in
1019:cylindrical
882:determinant
630:At a point
423:eigenvalues
173:measure of
160:hyperboloid
25:hyperboloid
4148:Categories
3462:References
2143:orthogonal
1198:'s theorem
1177:'s theorem
1025:of radius
896:definition
4063:Gauss map
3849:Curvature
3841:of curves
3827:curvature
3788:EMS Press
3647:CiteSeerX
3446:Gauss map
3393:Γ
3378:Γ
3374:−
3360:Γ
3345:Γ
3327:Γ
3312:Γ
3308:−
3294:Γ
3279:Γ
3261:Γ
3251:∂
3247:∂
3242:−
3228:Γ
3218:∂
3214:∂
3194:−
3147:π
3130:−
3117:π
3096:→
3037:π
3020:−
3014:π
2990:→
2941:σ
2938:Δ
2930:σ
2913:−
2838:∇
2597:−
2573:∇
2549:∇
2530:∇
2504:−
2359:−
2343:⋅
2244:∂
2240:∂
2204:∂
2200:∂
2168:−
2103:−
1956:−
1881:−
1844:−
1784:−
1742:−
1675:−
1654:−
1530:≠
1479:
1456:−
1437:∫
1421:ψ
1398:
1377:ϕ
1345:ψ
1330:
1315:ϕ
1300:
1285:ϕ
1222:umbilical
1220:that non-
1131:χ
1128:π
1097:∂
1093:∫
1070:∫
921:isometric
917:invariant
914:intrinsic
812:∬
805:π
793:θ
772:∑
763:radians.
526:∇
516:∇
512:−
503:∇
493:∇
200:in 1827.
175:curvature
171:intrinsic
113:κ
103:κ
4169:Surfaces
4132:Category
4005:Holonomy
3594:(1952).
3425:See also
1502:′
1486:′
1196:Liebmann
974: :
940:embedded
868:Gauss's
567:⟩
484:⟨
421:are the
413:The two
179:embedded
29:cylinder
3790:, 2001
2140:For an
1713:(after
1572:of the
1214:spheres
1175:Minding
884:of the
700:is the
623:is the
615:is the
425:of the
419:surface
388:Spheres
155:
141:
64:surface
3759:
3740:
3711:
3649:
3606:
3570:
3542:
3512:
3487:
1574:second
1492:
1413:, and
1369:where
1023:sphere
696:where
595:where
326:< 0
307:> 0
263:. The
51:, the
33:sphere
3726:Books
3068:of a
2962:of a
2467:) = 0
1578:first
966:is a
942:into
164:torus
3757:ISBN
3738:ISBN
3709:ISBN
3604:ISBN
3568:ISBN
3540:ISBN
3510:ISBN
3485:ISBN
3066:area
2877:and
2476:and
1705:The
1584:and
1576:and
1515:(an
962:. A
731:The
619:and
79:and
3657:doi
3089:lim
2983:lim
2899:):
2875:= 0
2152:= 0
1625:det
1605:det
1549:.
1470:sin
1395:cos
1327:sin
1297:cos
956:in
902:in
667:det
603:= â
574:det
453:of
353:= 0
55:or
47:In
4150::
3786:,
3780:,
3684:^
3655:.
3643:22
3641:.
3590:;
3397:22
3382:11
3364:12
3349:12
3331:12
3316:11
3298:11
3283:12
3265:11
3232:12
3180::
3111:12
2886:=
2882:=
2490::
2299:=
1588::
1582:II
994:â©
978:â
847:.
715:.
704:.
627:.
405:.
394:.
386:.
379:.
363:.
286:.
272:=
254:,
187:.
166:.
70:,
35:).
4039:e
4032:t
4025:v
3818:e
3811:t
3804:v
3765:.
3746:.
3717:.
3663:.
3659::
3612:.
3550:.
3548:.
3518:.
3493:.
3408:)
3402:2
3387:2
3369:2
3354:2
3341:+
3336:2
3321:1
3303:2
3288:1
3275:+
3270:2
3254:v
3237:2
3221:u
3208:(
3202:E
3199:1
3191:=
3188:K
3155:4
3151:r
3142:)
3139:r
3136:(
3133:A
3125:2
3121:r
3104:+
3100:0
3093:r
3085:=
3082:K
3045:3
3041:r
3032:)
3029:r
3026:(
3023:C
3017:r
3011:2
3005:3
2998:+
2994:0
2987:r
2979:=
2976:K
2944:.
2926:e
2922:2
2918:1
2910:=
2907:K
2893:Î
2888:e
2884:G
2880:E
2873:F
2851:4
2846:|
2841:F
2834:|
2828:|
2822:0
2815:z
2811:F
2803:y
2799:F
2791:x
2787:F
2777:z
2773:F
2765:z
2762:z
2758:F
2750:z
2747:y
2743:F
2735:z
2732:x
2728:F
2718:y
2714:F
2706:z
2703:y
2699:F
2691:y
2688:y
2684:F
2676:y
2673:x
2669:F
2659:x
2655:F
2647:z
2644:x
2640:F
2632:y
2629:x
2625:F
2617:x
2614:x
2610:F
2603:|
2594:=
2586:4
2581:|
2576:F
2569:|
2563:|
2557:0
2552:F
2539:T
2534:F
2525:)
2522:F
2519:(
2516:H
2510:|
2501:=
2498:K
2488:)
2486:F
2484:(
2482:H
2473:F
2471:â
2465:z
2463:,
2461:y
2459:,
2457:x
2455:(
2453:F
2432:2
2427:)
2421:2
2416:y
2412:F
2408:+
2403:2
2398:x
2394:F
2390:+
2387:1
2383:(
2375:2
2370:y
2367:x
2363:F
2354:y
2351:y
2347:F
2338:x
2335:x
2331:F
2324:=
2321:K
2311:)
2309:y
2307:,
2305:x
2303:(
2301:F
2297:z
2280:.
2276:)
2269:G
2266:E
2260:v
2256:E
2247:v
2235:+
2229:G
2226:E
2220:u
2216:G
2207:u
2194:(
2185:G
2182:E
2177:2
2173:1
2165:=
2162:K
2150:F
2147:(
2122:2
2117:)
2111:2
2107:F
2100:G
2097:E
2093:(
2085:|
2079:G
2074:F
2067:u
2063:G
2057:2
2054:1
2045:F
2040:E
2033:v
2029:E
2023:2
2020:1
2009:u
2005:G
1999:2
1996:1
1987:v
1983:E
1977:2
1974:1
1967:0
1961:|
1951:|
1945:G
1940:F
1933:v
1929:G
1923:2
1920:1
1911:F
1906:E
1899:u
1895:G
1889:2
1886:1
1876:v
1872:F
1862:v
1858:E
1852:2
1849:1
1839:u
1835:F
1827:u
1823:E
1817:2
1814:1
1805:u
1802:u
1798:G
1792:2
1789:1
1779:v
1776:u
1772:F
1768:+
1763:v
1760:v
1756:E
1750:2
1747:1
1736:|
1728:=
1725:K
1691:.
1683:2
1679:F
1672:G
1669:E
1662:2
1658:M
1651:N
1648:L
1642:=
1636:)
1632:I
1628:(
1620:)
1616:I
1612:I
1608:(
1599:=
1596:K
1586:I
1565:R
1533:1
1527:C
1499:v
1495:d
1483:v
1474:2
1464:2
1460:C
1453:1
1446:v
1441:0
1433:=
1430:)
1427:v
1424:(
1401:v
1392:C
1389:=
1386:)
1383:v
1380:(
1357:)
1354:)
1351:v
1348:(
1342:,
1339:)
1336:u
1333:(
1324:)
1321:v
1318:(
1312:,
1309:)
1306:u
1303:(
1294:)
1291:v
1288:(
1282:(
1259:C
1253:R
1247:C
1241:R
1235:C
1209:R
1203:C
1181:K
1143:,
1140:)
1137:M
1134:(
1125:2
1122:=
1119:s
1116:d
1110:g
1106:k
1100:M
1089:+
1086:A
1083:d
1079:K
1074:M
1032:R
1027:R
1010:R
996:U
992:S
986:R
980:V
976:U
972:f
959:R
954:S
945:R
905:R
900:S
891:R
831:.
828:A
825:d
821:K
816:T
808:+
802:=
797:i
787:3
782:1
779:=
776:i
761:Ï
753:Ï
749:Ï
745:Ï
698:S
684:,
681:)
677:p
673:(
670:S
664:=
661:)
657:p
653:(
650:K
639:R
633:p
621:g
610:i
606:e
600:i
597:â
583:,
577:g
560:2
555:e
550:,
545:1
540:e
535:)
530:2
520:1
507:1
497:2
489:(
476:=
473:K
455:f
447:p
443:f
439:p
435:f
351:2
348:Îș
345:1
342:Îș
324:2
321:Îș
318:1
315:Îș
305:2
302:Îș
299:1
296:Îș
283:2
280:Îș
277:1
274:Îș
270:Î
260:2
257:Îș
251:1
248:Îș
151:r
147:/
144:1
136:r
122:.
117:2
107:1
99:=
96:K
85:2
82:Îș
76:1
73:Îș
60:Î
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